Connecting two planes together successfully
After removing doubles (select all > W > Remove Doubles) and recalculating normals (select all > Ctrl+N) only this part of geometry is displayed as non-manifold:



When inspecting one of those sides it becomes clear that vertices of thin part of mesh aren't connected to thick part:



You should connect those vertices to edge located between thick and thin parts. 


Select top and bottom vertices of that edge. Subdivide them with W > Subdivide, in Last Operator panel (F6) choose 2 subdiviosions.


Select top subdivided vertex, select top original vertex and press Alt+M > At Last to merge them.


Repeat that for 3 other vertices (to do that faster you can use Repeat Last operator. Once you merged first pair of vertices select next pair appropriately and press Shift+R. It will repeat last action).


Optionally: you could edit topology a bit to avoid Ngons on thick part of this mesh. However this shouldn't be important if you aren't going to add Subsurf on the model.